general question regarding roots and proportions--triangles

[wignewton]

I have a question on how to get one of the legs of a triangles measurements using the 30.60.90 proportion method and the known leg size. for example, if the known leg size of the 60 side is 12, how do you get the 30 side? I know you need to divide by sqr. 3. is there an easy way to do this? does the question make sense? thanks--dan

[Ben Ku]

First, the derivation: You have to draw this along with me. :)

Imagine an equilateral triangle ABC with sides of length 2. Draw an altitude from the top vertex A to the base BC at point D. What we've done is divide this equilateral triangle into two 30-60-90 right triangles: triangles ABD and ACD. Because this is an equilateral triangle, angles B and C have measures 60 degrees, and angles BAD and CAD are 30 degrees.

So in the right triangle ABD, AB is the hypotenuse with length 2, BD is one leg with length 1 (half the hypotenuse), and the other leg (AD) can be found to sqrt(3). So the ratio of the sides is

short leg : long leg : hypotenuse = 1 : sqrt (3) : 2.


Now, how can we use this: Always get to the short leg.
If you have the hypotenuse, divide by 2 to get to the short leg, then multiply by sqrt(3) for the long leg. If the hypotenuse is 10, the short leg is 5, and the long leg is 5sqrt(3).

If the long leg is 6sqrt(3), the short leg is 6, and the hypotenuse is 12.

I hope this helps.